The singular perturbation of surface tension in Hele-Shaw flows
Morphological instabilities are common to pattern formation problems such as the non-equilibrium growth of crystals and directional solidification. Very small perturbations caused by noise originate convoluted interfacial patterns when surface tension is small. The generic mechanisms in the formation of these complex patterns are present in the simpler problem of a Hele-Shaw interface. Amid this extreme noise sensitivity, what is then the role played by small surface tension in the dynamic formation and selection of these patterns? What is the asymptotic behaviour of the interface in the limit as surface tension tends to zero? The ill-posedness of the zero-surface-tension problem and the singular nature of surface tension pose challenging difficulties in the investigation of these questions. Here, we design a novel numerical method that greatly reduces the impact of noise, and allows us to accurately capture and identify the singular contributions of extremely small surface tensions. The numerical method combines the use of a compact interface parametrization, a rescaling of the governing equations, and very high precision. Our numerical results demonstrate clearly that the zero-surface-tension limit is indeed singular. The impact of a surface-tension-induced complex singularity is revealed in detail. The singular effects of surface tension are first felt at the tip of the interface and subsequently spread around it. The numerical simulations also indicate that surface tension defines a length scale in the fingers developing in a later stage of the interface evolution.
"Reprinted with the permission of Cambridge University Press." Received April 4 1999, Revised September 15 1999, Published Online 08 September 2000 We would like to thank Saleh Tanveer and Mark Kunka for teaching us much about the daughter singularity and the asymptotic theory. We also benefitted greatly from the advice of Pingwen Zhang and Helen Si about the numerical aspects of this work. This research is supported in part by a ONR grant N00014-96-1-0438 and by a NSF grant DMS-9704976.